The article provides a summary of the elementary ideas about vectors usually met in school mathematics, describes what vectors are and how to add, subtract and multiply them by scalars and indicates why they are useful.

Vector Walk

Stage: 4 and 5 Challenge Level:

Why do this problem?

This problem encourages students to think about vectors as representing a movement from one point to another. The need for coordinate representation of points will emerge automatically and the problem naturally requires an interplay between geometry and algebra.

Possible approach

Set students the challenge to investigate possible end points when combining steps of vectors $b_1$ and $b_2$ in a vector walk. Some students will prefer to work algebraically while others will wish to represent the problem geometrically; by encouraging students to work in groups with others who have different preferred methods, rich mathematical thinking can emerge.

Students should aim to describe geometrically the set of points which can be made by combining the two vectors.

Once students have successfully described the set of points made from combinations of $b_1$ and $b_2$, set them the two challenges - to find other pairs of basic vectors which yield the same possibilities, and to find a pair of basic vectors which will never lead to the points found in the first part of the question.

Key questions

What do the points you can reach with $b_1$ and $b_2$ have in common?

Can you describe the resulting set of points geometrically (i.e. describe them clearly without algebra)?

Possible extension

Polygon Walk explores vector walks which form polygons around the origin.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities
can be found here.